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92                        ANALYTICAL MECHANICS
The last equation cannot be true unless
f =dx.
T*     dt
and                        f,= f-
Therefore the component of the acceleration along a fixed line equals the time rate of change of the component of the velocity along that line. It follows from the last two equations that:
, _ dx _       _ .-dt     dt       '
v~ dt     dt
The magnitude and the direction of the acceleration are: given by the following equations :
~                                 (X)
where 0 is the angle f makes with the re-axis.
89. Tangential and Normal Components of Acceleration.  The tangential component of the acceleration at P (Fig. 57) equals the rate at which the velocity increases along the direction of the tangent at P. In order to find this rate we consider the velocities at two neighboring points PI and P2. Let YI and v2 be the velocities at these points and i and e2 the angles which YI and v2 make with the tangent at P. Then the change in the velocity along the tangent at P, while the particle moves from PI to P2, is
Vz COS 2  Vi COS i.
Dividing this by the corresponding interval of time we obtain the average rate at which the velocity increases from